If $X$ is a complex Abelian variety of dimension $g$, then

- The canonical sheaf is trivial
- $\dim {\rm H}^i(X; \mathcal{O}_X) = \binom{g}{i}$.

When $g =1,2$, then any connected, projective nonsingular $X$ satisfying the above two must be an Abelian variety. Is this true for higher $g$? If not, what other conditions can I add? Or is such a request unreasonable?

Disclaimer: I don't know very much about Abelian varieties, so apologies if this material is standard. A search in the literature turned up some papers about characterizations of Abelian varieties up to birational equivalence, but under weaker assumptions. I really want to know if I'm given a variety $X$, how to tell if it is Abelian or not via some sort of reasonably accessible sheaf-related conditions. I'm most interested in the case $g=3$, but results for other $g$ are welcome also.